Rheology describes the relation between the strain or rate of strain field and the stress field. In simple flows, viscosity is a single parameter that links the rate of shear and the shear stress in the flow field. However, in most real, industrial fluids, where the fluids are multi-phase and complex (solid-liquid dispersions and suspensions), the viscosity cannot be represented in terms of a single parameter and becomes a function of the flow field. It is well known that in a solid-liquid slurry, the local fluid viscosity not only depends on the local concentration of the solids but also on the local rate of shear and its gradient. Often, the solids being transported in the pipeline migrate away from the solid walls and into the core of the flow. As a result, measurement of rheology of the fluid near the wall will yield erroneous results relative to the total flow cross section.
Rheological characterization of solid-liquid dispersions is commonly performed using off-line measurement devices. This approach has the disadvantage that once a sample is withdrawn from the process stream its rheological properties will begin to change. Most often, the fluids to be characterized have rheologies that intimately depend on the flow field. This dependence is especially true for colloidal suspensions in which size and fractal dimensions of the clusters or aggregates depend strongly on the environment under which they exist. Many of these fluids exhibit shear-dependent viscosity, in the form of shear thinning or shear-thickening behavior, requiring determination of their viscosity at various shear-rates which correspond to the range of shear rates observed in the flow field. Off-line measurements can hardly reproduce the same conditions which exist in a real flow field such as shear induced migration of solid particles. Further, given that the material in the pipeline may not be homogeneous, it will be difficult to obtain a representative sample for off-line measurements.
Existing real-time on-line process monitoring rheometers monitor the properties of a side stream of material (Dealy, J. M. And Wissbrun, K. F., 1990, "Melt Rheology and Its Role in Plastics Processing," Van Nostrand Reinhold, New York). The steady shear viscosity is measured at a single shear rate (or flow rate). To obtain viscosity at various shear rates, either the flow rate in the side stream is controlled by an auxiliary pump or several parallel or serial side streams at different flow velocities are produced. For example, the current capillary viscometry technology uses an auxiliary pumping capacity and by cascading a series of capillary tubes, which limits the number of data points to the number of tubes, provides multiple-point viscosity measurements.
Commonly used for measurement of liquid carrier flow rates, ultrasonic Doppler flowmeters can yield the mean velocity inside of a pipe or in a flow field (Fowlis, W. W., 1973, "Liquid Metal Flow Measurements Using an Ultrasonic Doppler Velocimeter," Nature Phys. Sci., 242, pp. 12-13). This approach relies on the principle that the frequency of a longitudinal acoustic wave, reflected from discontinuities or scatterers moving with the flow stream, is shifted. FIG. 1 depicts the most common orientation of the transmitter 100 and receiver 102 probes with respect to a pipe 104 and a fluid 106 flowing therein. The difference between the transmitted and received frequencies (frequency shift) is directly proportional to the discontinuity or scatterer velocity. ##EQU1## where v is the convection velocity, c is the speed of sound, and .theta. is the half angle between the transmitted and received beams of ultrasonic energy. By assuming that the scatterers move at the same velocity as the fluid, one obtains the flow velocity. Numerous papers have been written on this topic, dating back to the U.S. patent to Chilowski, C. and Langevin, P., 1923, "Production of Submarine Signals and the Location of Submarine Objects," U.S. Pat. No. 1,471,547. Today, this device could yield accuracies on the order of 1% of full scale. An estimated ten thousand or more of ultrasonic Doppler flowmeters are currently being used in the process industries such as pulp and paper, minerals, and power industries. Comprehensive reviews of the commercially available ultrasonic Doppler flowmeters are provided in Lynnworth, L. C., 1989, "Ultrasonic Measurements for Process Control--Theory, Techniques, Applications," Academic Press, San Diego, Calif. Lynnworth (1989), and Asher, R. C., 1983, "Ultrasonic Sensors in the Chemical and Process Industries," J. Phys. E. Sci. Instrum., 16, pp. 959-63.
Reflection-mode and transmission-mode ultrasonic tomographic imaging systems have been well developed during the past few decades. A thorough review of these systems is provided by Plaskowski, A., Beck, M. S., Thorn, R., and Dyalowski, T., 1995, "Imaging Industrial Flows," IOP Publishing, Ltd, Bristol. The use of transmission-mode ultrasonic tomography for measuring the three dimensional velocity profile in a pipe has been demonstrated by Johnson, S. A., Greenleaf, J. F., Hansen, C. R., Tanaka, W. F., Lent, A., Christensen, D. A., and Woolley, R. L., 1977, "Reconstructing Three-Dimensional Fluid Velocity Vector Fields from Acoustic Transmission Measurements," Acoustical Holography, 7, L. W. Kessler Ed., Plenum Press, New York, and Hildebrand, B. P. and Liem, R., 1985, "Ultrasonic Tomography for Measuring Mass Flow Rates and for Mapping Spatial Distribution of Fluid Flows," Final Report for DOE Contract DE-AC03-84ER80150. Most of the systems are based on a transmission-mode fan beam projection in which the signal transmitted by a transducer is received by several transducers on the opposite side of the pipe. Due to the serious refractive effects associated with acoustic transmission through a metal pipe at off-normal views (off diametrical axis), parallel projections, such as those obtained with x-ray and .gamma.-rays, have been shown to be impractical for true non-intrusive ultrasonic techniques (i.e., transducers located outside the pipe wall as in the case of clamp-on devices.). However, drill-thru flush-mounted transducer installations have been shown to produce fairly robust signatures in metal pipes with very little effect on the flow field as reported by Lynnworth, L. C., 1989, "Ultrasonic Measurements for Process Control--Theory, Techniques, Applications," Academic Press, San Diego, Calif.
In a tomographic imaging system, the object space is viewed from several parallel projections. These projections are essentially a series of time delay measurements in an acoustic system, giving the distances between the object-media interfaces from the receiving transducers. In the reflection-mode (pulse-echo), each transducer is used to transmit a short pulse and then receive the resulting reflected acoustic energy as a function of time. Unlike electrical resistive or capacitive impedance sensing technique, acoustic reconstruction algorithms are linear and straightforward. The output of the reconstruction process provides the echo strength versus time delay. The strength of the echo is a function of the sector angle aperture, local gradient of concentration of the slurry (gradient of acoustic impedance), and opacity of the mixture along the pathlength of the beam if the mixture is attenuative. The time delay axis may be converted to a distance (range) from the knowledge of local acoustic velocity. ##EQU2## where c is the local acoustic velocity or local speed of sound and X is the range. It has been assumed that the local speed of sound is a constant over the entire measurement range, thereby the range is directly proportional to the range time delay. The average speed of sound is found from two opposite transducers placed across the pipe. Thus, the variance in measurement of the range directly depends on the variance in the local speed of sound or concentration of the medium. If the local concentration of the slurry changes dramatically from one point to the next in the flow field, then the range will have a substantial amount of variance and the resulting measurement will be inaccurate. Although data exists for monodisperse particle slurries which show the dependence of the speed of sound on the slurry concentration (Kytomaa, H. K., 1995, "Theory of Sound Propagation in Suspensions: A Guide to Particle Size and Concentration Characterization," Powder Technology, 82, pp. 115-121), the actual dependence of speed of sound on concentration would have to be determined for the particular slurry in question using off-line calibration.
In the transmission-mode (time-of-flight) tomography, an ultrasonic wave is transmitted by one transducer and received on the opposite side by a receiving transducer. In the case of an invariant velocity field, the time-of-flight from the transmitter to receiver is ##EQU3## For a discretized field, the above equation turns into I rays and along each ray m cells. Thus a system of I linear equations are obtained. If the number of independent measurements through each cell becomes sufficiently large (&gt;m), then the equations may be solved for the unknown component c in each cell. Addition of a fluid velocity vector usually complicates the matter since the velocity of sound in each cell becomes directionally dependent. ##EQU4## where v is the convection velocity and s is the unit vector along the ray path. The scalar and vector effects may be separated out by calculating the sum and difference of measurements of time-of-flight taken from the same path. The resolution in the time-of-flight measurement is limited to the points at which different rays cross. As the number of projections (rays from different directions) increase, the resolution in velocity measurement improves.
In summary, the reflection mode or the pulse-echo technique is considered more suitable for the measurement of the convection of fluid velocity. However, speed of sound measurements may be more simply obtained using the transmission mode or the time-of-flight measurements, but at a lower spatial resolution.
Brunn, P. O., Vorewerk, J., and Steger, R., 1993, "Optical and acoustic rheometers: three examples," Rheology 93, March 1993, pp. 20-27; and Vorwek, J., Steger, R., Teufel, M., and Brunn, P. O., 1994, "Use of an optical meter to measure the flow rate and apparent viscosity of non-Newtonian fluids," Flow Meas. Instrum., Vol. 5, No. 1, pp. 51-57 discuss ultrasonic measurements of the local shear rate as determined from the measured local velocity in the pipe. This was achieved by using an ultrasonic reflection-mode (pulse-echo) Doppler velocity mapping system. The principle of operation of the system is as follows: Ultrasonic transmission time-of-flight measurements can be used to determine the integrated line-of-sight acoustic velocity in the fluid. If the fluid contains scatterers (e.g. particles), then a coherent reflection system can be used to measure the Doppler frequency shift caused by the fluid flow. The magnitude of the Doppler shift can be used to determine the fluid velocity. Applying a sequence of range gates to the Doppler measurements allows determination of the fluid velocity profile along the line-of-sight of the ultrasonic transducer. When the acoustic velocity is uniform across the cross-section of the pipe, and is axisymmetric then this range-gated ultrasonic Doppler data can provide accurate measurement of the fluid velocity profile.
In pipes involving flow of complex fluids of unknown or varying properties, both the fluid velocity profile and the acoustic velocity are non-uniform across the cross-section of the pipe. The time-of-flight acoustic transmission measurement will be in error because the fluid velocity profile is not taken into account, and the reflection Doppler measurement will be in error because the acoustic velocity profile is not known. This will cause a distortion of the fluid velocity profile measured by the ultrasonic Doppler system, because the Doppler shift is proportional to the acoustic velocity and the range gate mapping assumes a uniform acoustic velocity across the pipe. Thus, a single line measurement cannot accurately solve the complex pipe flow velocity problem. The transmission measurement can only solve for the average acoustic velocity, not the acoustic velocity profile. The reflection Doppler measurement requires the acoustic velocity profile in order to determine the actual fluid velocity profile. A single line-of-sight measurement system does not provide enough information to simultaneously solve for both the acoustic and the fluid velocity profiles.
Accordingly, there is a need in the field of acoustic fluid velocity measurement for a method and apparatus that provides an accurate acoustic and fluid velocity profiles with high spatial resolution for fluids having spatially varying speed of sound.